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Classification of the space spanned by theta series and applications
Authors:
YoungJu Choie and Olav K. Richter
Journal:
Proc. Amer. Math. Soc. 135 (2007), 2309-2315
MSC (2000):
Primary 14K25; Secondary 11F50, 11F03
Posted:
March 14, 2007
MathSciNet review:
2302551
Full-text PDF Free Access
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Abstract: We determine a class of functions spanned by theta series of higher degree. We give two applications: A simple proof of the inversion formula of such theta series and a classification of skew-holomorphic Jacobi forms.
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- A. Andrianov and G. Maloletkin, Behavior of theta series of degree
 under modular substitutions, Math. USSR-Izvestija 9 (1975), no. 2, 227-241.MR 0379382 (52:287)
- 2.
- -, Behavior of theta series of genus
 of indefinite quadratic forms under modular substitutions, Proc. Steklov Inst. Math. 4 (1980), 1-12.MR 0558935 (81d:10017)
- 3.
- T. Arakawa, Siegel's formula for Jacobi forms, Internat. J. Math. 4 (1993), no. 5, 689-719.MR 1245348 (95b:11047)
- 4.
- R. Bellman and S. Lehman, The reciprocity formula for multidimensional theta functions, Proc. Amer. Math. Soc. 12 (1961), 954-961.MR 0136775 (25:236)
- 5.
- Y. Choie and Y. Taguchi, A simple proof of the modular identity for theta series, Proc. Amer. Math. Soc. 133 (2005), no. 7, 1935-1939.MR 2137858 (2005m:11080)
- 6.
- W. Couwenberg, A simple proof of the modular identity for theta functions, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3305-3307.MR 1990617 (2004e:11040)
- 7.
- M. Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1966.MR 0209258 (35:160)
- 8.
- M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, 1985.MR 0781735 (86j:11043)
- 9.
- E. Freitag, Siegelsche Modulfunktionen, Springer, Berlin, Heidelberg, New York, 1983.MR 0871067 (88b:11027)
- 10.
- S. Friedberg, On theta functions associated to indefinite quadratic forms, J. Number Theory 23 (1986), 255-267. MR 0845907 (87k:11051)
- 11.
- S. Hayashida, Klingen type Eisenstein series of skew-holomorphic Jacobi forms, Comment. Math. Univ. St. Pauli 52 (2003), no. 2, 219-228.MR 2029023 (2004k:11070)
- 12.
- -, Skew-holomorphic Jacobi forms of index 1 and Siegel modular forms of half-integral weight, J. Number Theory 106 (2004), no. 2, 200-218.MR 2059071 (2005i:11066)
- 13.
- O. Richter, On transformation laws for theta functions, Rocky Mountain J. Math. 34 (2004), no. 4, 1473-1481. MR 2095586 (2005j:11032)
- 14.
- N-P. Skoruppa, Developments in the theory of Jacobi forms, Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk (1990), 167-185. MR 1096975 (92e:11043)
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- H. Stark, On the transformation formula for the symplectic theta function and applications, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 29 (1982), 1-12.MR 0657868 (83j:10029)
- 16.
- C. Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191-224.MR 1049896 (91g:11046)
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Additional Information
YoungJu Choie
Affiliation:
Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Korea
Email:
yjc@postech.ac.kr
Olav K. Richter
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203
Email:
richter@unt.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08706-0
PII:
S 0002-9939(07)08706-0
Received by editor(s):
August 4, 2004
Received by editor(s) in revised form:
March 9, 2006
Posted:
March 14, 2007
Additional Notes:
The first author was partially supported by KOSEF R01-2003-00011596-0 and ITRC
Communicated by:
Juha M. Heinonen
Article copyright:
© Copyright 2007 American Mathematical Society
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